In the late Winter of 1927, Neils Bohr went skiing for a few weeks in Norway, during which he analyzed the puzzling situation in quantum mechanics in deeply philosophical terms.

In the previous two years, Max Born, with his clever students Werner Heisenberg and Pascual Jordan, had developed the quantum mechanics of material particles. They had derived most of the results of Bohr's old quantum theory, eliminating his idea of semi-classical orbits but confirming Bohr's "quantum postulate of stationary states with electrons "jumping" between them, radiating energy with E2 - E1 = hν, following Max Planck's hypothesis about the quantum of action.

And just the year before, Erwin Schrödinger developed an alternative "wave mechanics," which he showed gives exactly the same results as quantum mechanics, but without some of the major assumptions in Bohr's earlier work, which had been adopted also by Heisenberg. In his 1929 textbook, Heisenberg dubbed their work "Der Kopenhagener Geist," many years later known as the "Copenhagen interpretation" of quantum mechanics. Where Bohr and Heisenberg described the stationary states with arbitrary quantum numbers, Schrödinger showed quantum numbers emerge naturally from the number of nodes in his wave function that could fit around an electron orbit (an idea that Louis de Broglie had proposed earlier).

The dualistic view that matter might consist of either particles or waves (or maybe both) must surely have inspired Bohr to think about complementary relations, but there are strong reasons to think that he might not have wanted to identify his complementarity with Einstein's ideas about "wave-particle duality".

Heisenberg said that "The main point was that Bohr wanted to take this dualism between waves and corpuscles as the central point of the problem." But Bohr also used the term complementary to describe the "reciprocal uncertainty" between momentum and position in Heisenberg's indeterminacy relations. Bohr said:

the measurement of the positional coordinates of a particle is accompanied not only by a finite change in the dynamical variables, but also the fixation of its position means a complete rupture in the causal description of its dynamical behaviour, while the determination of its momentum always implies a gap in the knowledge of its spatial propagation. Just this situation brings out most strikingly the complementary character of the description of atomic phenomena [italics added]

Bohr may never have completely accepted Albert Einstein's idea that light itself might consist of particles, since quantum particles are complements of classical waves. In 1905, Einstein had proposed his "light-quantum hypothesis," that light came in discrete and discontinuous quantities, something like Newton's "light corpuscles."

Einstein wrote in 1905:

On the modern quantum view, what spreads out is a wave of probability amplitude for absorbing a whole "light quantum" somewhere. The wave function ψ should be thought of as a "possibility" function

In accordance with the assumption to be considered here, the energy of a light ray spreading
out from a point source is not continuously
distributed over an increasing space but consists
of a finite number of energy quanta which are
localized at points in space, which move without
dividing, and which can only be produced and
absorbed as whole units.

Bohr resisted Einstein's "light-quantum hypothesis" in 1913. His Bohr model of the atom postulated that there are "stationary states" with energy levels En. His second postulate was that electrons jump discontinuously between levels, emitting or absorbing radiation of frequency ν, where

Em - En = hν

As obvious as it is today that Bohr's hν is a "photon" (as it was dubbed in the middle 1920's), Bohr thought that the radiation emitted or absorbed was continuous and classical electromagnetism. It is not clear that Bohr had completely accepted photons and the dual nature of light even as he formulated his philosophical notion of complementarity in his "Como Lecture" of 1927. He seems to have accepted it in 1949, in his tribute to Einstein.

Einstein had written as early as 1909 that the wave theory of light might need to be augmented to explain his particle-like properties.

This was the beginning of wave-particle duality that Bohr would reconcile with the idea of complementarity in quantum mechanics

When light was shown to exhibit interference and diffraction, it seemed almost certain that light should be considered a wave...A large body of facts shows undeniably that light has certain fundamental properties that are better explained by Newton's emission theory of light than by the oscillation theory. For this reason, I believe that the next phase in the development of theoretical physics will bring us a theory of light that can be considered a fusion of the oscillation and emission theories...

("On the Development of Our Views Concerning the Nature and Constitution of Radiation," Physikalische Zeitschrift, 10, p.817)

When Bohr returned from his skiing vacation, he received a draft paper from Heisenberg claiming that some physical variables might be measured precisely, but then their canonically conjugate variables would have a very large error. This is his famous "indeterminacy principle." If a momentum measurement has accuracy Δp and position accuracy Δx than the product of the two indeterminacies is Δp Δx ≥ h, where h is Planck's constant for the quantum of action.

Bohr asked Heisenberg to include his notion of complementarity, and perhaps his derivation of indeterminacy from pure wave-mechanical considerations, in his new paper. This upset Heisenberg greatly, because he thought that Schrödinger's "wave mechanics" added nothing to his particle-oriented "matrix mechanics." Bohr thought both were needed. Though somewhat contradictory, they were his first example of "complementarity."

Definitions of complementarity today almost always include wave-particle duality, but Bohr was so vague about the precise meaning of his term complementarity when he introduced it in his 1927 "Como Lecture" that it is confusing to this day. One thing he did in the Como Lecture was to argue that both Heisenberg's discontinuous and indeterministic particle picture and Schrödinger's continuous and deterministic wave picture were both needed in quantum mechanics. The theories themselves, matrix mechanics and wave mechanics, are "complementary."

Almost no one, least of all Bohr, gave credit to Einstein, for his 1909 insight that both wave and particle pictures needed to be fused, or to his views in the early 1920's that the wave was a "Gespensterfeld" (ghost field) that guides the particles. Ironically, and unjustly, to this day the "Bohr atom" is taught as discontinuous "jumps" between energy levels accompanied by the emission or absorption of a photon, whereas Bohr fought against Einstein's light quantum hypothesis for decades. Einstein developed the quantum theory of radiation, explaining emission, absorption, and the radical hypothesis of "stimulated emission" (that led to the invention of the laser) in 1916! But it is Bohr's name most often cited.

Bohr claimed that an experimental apparatus must always be treated as a classical object and described using ordinary language. He thought that specific experiments could reveal only part of the quantum nature of microscopic objects. For example, one experiment might reveal a particle's dynamical properties such as energy, momentum, position, etc. Another experiment might reveal wavelike properties. But no one experiment could exhaustively reveal both. The experiments needed to reveal both are "complementary."

Bohr's first definition of complementarity in the Como lecture somewhat opaquely contrasts the "space-time coordination" with the "claim of causality."

Space-time co-ordination and the claim of causality are complementary.

They "symbolize" observationand definition, also complementary?

Relativity has a limit v / c → 0.

Quantum mechanics has the limit h → 0(better h / m → 0).

The very nature of the quantum theory thus forces us to regard the space-time co-ordination and the claim of causality, the union of which characterises the classical theories, as complementary but exclusive features of the description, symbolising the idealisation of observation and definition respectively. Just as the relativity theory has taught us that the convenience of distinguishing sharply between space and time rests solely on the smallness of the velocities ordinarily met with compared to the velocity of light, we learn from the quantum theory that the appropriateness of our usual causal space-time description depends entirely upon the small value of the quantum of action as compared to the actions involved in ordinary sense perceptions. Indeed, in the description of atomic phenomena, the quantum postulate presents us with the task of developing a 'complementarity' theory the consistency of which can be judged only by weighing the possibilities of definition and observation.

("The Quantum Postulate and the
Recent Development of Atomic Theory," Supplement to Nature, April 14, 1928, p.580)

And again, a few paragraphs later, Bohr looks for a complementary relation between the "kinematics" of a space-time picture and the "dynamics" of a causal picture using variables like momentum, energy, etc. :

This situation would seem clearly to indicate the impossibility of a causal space-time description of the light phenomena. On one hand, in attempting to trace the laws of the time-spatial propagation of light according to the quantum postulate, we are confined to statistical considerations. On the other hand, the fulfilment of the claim of causality for the individual light processes, characterised by the quantum of action, entails a renunciation as regards the space-time description.

Once again, space-time and causality are complementary views of classical concepts.

Of course, there can be no question of a quite independent application of the ideas of space and time and of causality. The two views of the nature of light are rather to be considered as different attempts at an interpretation of experimental evidence in which the limitation of the classical concepts is expressed in complementary ways.

("The Quantum Postulate and the
Recent Development of Atomic Theory," Supplement to Nature, April 14, 1928, pp.580-581)

Bohr points out that in expressions like ΔE Δt = h and Δp Δx = h, we see both space-time (wave) variables x, t and dynamical (particle) variables E, p.

As mentioned above, Bohr thought Heisenberg's "uncertainty" could be an example of complementarity, because two different measurement apparatuses were needed to measure dynamical momentum and space-time position.

An important contribution to the problem of a consistent application of these methods has been made lately by Heisenberg (Zeitschr. f. Phys., 43, 172; 1927). In particular, he has stressed the peculiar reciprocal uncertainty which affects all measurements of atomic quantities. Before we enter upon his results it will be advantageous to show how the complementary nature of the description appearing in this uncertainty is unavoidable already in an analysis of the most elementary concepts employed in interpreting experience.

("The Quantum Postulate and the
Recent Development of Atomic Theory," Supplement to Nature, April 14, 1928, p.581)

Bohr notes that Heisenberg's derivation of his indeterminacy principle was entirely done with particles and dynamical variables. Bohr then proceeds to derive Heisenberg's relations solely on the basis of a wave theory (a space-time description). This must have embarrassed Heisenberg, who resisted at first but eventually completely accepted and promoted Bohr's view of complementarity as an essential part of the Copenhagen Interpretation (along with his own uncertainty principle and Born's statistical interpretation of the wave function).

The use of a wave description reduces sharpness in definitions

Here the complementary
character of the description appears,
since the use of wave-groups is necessarily accompanied
by a lack of sharpness in the definition of
period and wave-length, and hence also in the definition
of the corresponding energy and momentum
as given by relation (1).

We can illustrate Bohr's argument on lack of sharpness as a simple consequence of instrumental resolving power.

Δt is the time it takes the wave packet to pass a certain point. Δν is the range of frequencies of the superposed waves.

In space instead of time, the wave packet is length Δxand the range of waves per centimeter is Δσ.

Rigorously speaking, a limited wave-field can
only be obtained by the superposition of a manifold
of elementary waves corresponding to all values
of ν and σx, σy, σz. But the order of magnitude of
the mean difference between these values for two
elementary waves in the group is given in the most
favourable case by the condition

Δt Δν = Δx Δσx = Δy Δσy = Δz Δσz = 1,

where Δt, Δx, Δy, Δz denote the extension of the
wave-field in time and in the directions of space
corresponding to the co-ordinate axes. These
relations — well known from the theory of optical
instruments, especially from Rayleigh's investigation
of the resolving power of spectral apparatus
— express the condition that the wave-trains
extinguish each other by interference at the
space-time boundary of the wave-field. They
may be regarded also as signifying that the group
as a whole has no phase in the same sense as the
elementary waves. From equation (1) we find
thus:

Δt ΔE = Δx ΔIx = Δy ΔIy = Δz ΔIz = h, . . (2)

as determining the highest possible accuracy in
the definition of the energy and momentum of the
individuals associated with the wave-field. In
general, the conditions for attributing an energy
and a momentum value to a wave-field by means
of formula (1) are much less favourable. Even
if the composition of the wave-group corresponds
in the beginning to the relations (2), it will in the
course of time be subject to such changes that it
becomes less and less suitable for representing an
individual. It is this very circumstance which
gives rise to the paradoxical character of the
problem of the nature of light and of material
particles. The limitation in the classical concepts
expressed through relation (2) is, besides, closely
connected with the limited validity of classical
mechanics, which in the wave theory of matter
corresponds to the geometrical optics, in which
the propagation of waves is depicted through
'rays.' Only in this limit can energy and momentum
be unambiguously defined on the basis
of space-time pictures. For a general definition
of these concepts we are confined to the conservation
laws, the rational formulation of which has
been a fundamental problem for the symbolical
methods to be mentioned below.

In the language of the relativity theory, the
content of the relations (2) may be summarised in
the statement that according to the quantum
theory a general reciprocal relation exists between
the maximum sharpness of definition of the space-time
and energy-momentum vectors associated
with the individuals.

Bohr may still hope to "reconcile" conservation laws by claiming space-time points are "unsharp" (reminiscent of his BKS statistical conservation ideas).

This circumstance may be
regarded as a simple symbolical expression for the
complementary nature of the space-time description
and the claims of causality. At the same time,
however, the general character of this relation
makes it possible to a certain extent to reconcile
the conservation laws with the space-time coordination
of observations, the idea of a coincidence
of well-defined events in a space-time point being
replaced by that of unsharply defined individuals
within finite space-time regions.

("The Quantum Postulate and the
Recent Development of Atomic Theory," Supplement to Nature, April 14, 1928, pp.581-582)

To summarize, Bohr saw many elements of the new quantum mechanics as revealing his deep insight into complementarity. Among them were:

wave-particle duality was probably the proximate trigger, but Kant's noumena/phenomena was likely the original inspiration. And Bohr avoided referring to Einstein's years of work on wave-particle duality.

wave mechanics and particle/matrix mechanics as equally "true"

the indeterminacy principle, i.e., the reciprocal nature of the conjugate variables, momentum/position, energy/time, and action-angle

wave-packet limits on resolving power versus the disturbing effect of light on an observation

quantum systems, but apparatus described classically

all quantum evidence must be expressed in classical terms, "results of observations must be expressed in unambiguous language using terminology from classical physics," Heisenberg called this a paradox